Alphabeta MathPeriodic fluid Fourier Sobolev spaces
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Areas of Elementary Plane Figures
- Banach Valued Integration and the Radon Nikodym Property
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Lp Spaces and Test-Function Conventions
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convergence: Nets and Filters
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Density Separability and Convolution in Lᵖ
- Determinants of Matrices over a Commutative Ring
- Distributions Test Functions and Differentiation
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Function Space Topologies and the Exponential Law
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Hereditary and Productive Behaviour of the Separation Axioms
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Modes of Convergence Egorov and Lusin
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Orthonormal Bases, Parseval and Fourier Series
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Schwartz Space and the Plancherel Theorem
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Analytic Hahn Banach Theorem
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Weak Derivatives and Sobolev Spaces
2 · Logical flowchart
3 · Items
Definition
Periodic fluid Fourier Sobolev spaces
Definition
Projected no-slip Stokes Hilbert space
Lemma
Rectangular continuous divergence inverse on mean-zero tests
Lemma
Periodic Fourier embeddings and products
Definition
Periodic Leray and pressure maps
Lemma
Tame transport commutator
Theorem
Pressure recovery for periodic incompressible Sobolev velocity
Lemma
Directional Poincaré bound for zero trace
Theorem
Projected stationary weak Stokes solvability
Lemma
Difference energy for given smooth incompressible and symmetric solutions
Theorem
Rectangular no-slip Stokes velocity and distribution pressure
Sources
- Gerald Teschl, Topics in Real and Functional Analysis (2017)
- Theo Bühler and Dietmar Salamon, Functional Analysis
- Martin Vohralík, A posteriori error estimates for efficiency and error control in numerical simulations, Course NM497 lecture notes (2010–2011)
- Erick Schulz, Introduction to the Navier–Stokes Equations, McGill MATH 595 notes (2014)
- Terence Tao, 254A Notes 1: Local well-posedness of the Navier–Stokes equations (2018)